Consist of an amplifier A, and a frequency selective network β
$$A_{cl}(jω)=\frac{v_o}{v_i}=\frac{A}{1-Aβ(jω)}$$
For oscillation to occur:
$$Aβ=1\angle 0^{\circ}; |Aβ|=1, : A_{cl}(jω)\rightarrow\infty$$
2 aspects of oscillators that we consider:
Frequency of oscillation (ω0/f0)
Gainloss (amplitude of β); to decide amplifier gain s.t. |Aβ|=1
Wien bridge
Wien Bridge Oscillator
V2 on diagram: use a potential divider s.t.
$$v_2=\frac{Z_2}{Z_1+Z_2}, Z_1=R+\frac{1}{sC}, Z_2=R//\frac{1}{sC}=\frac{R}{1+sCR}$$
letting s=jω. v2 is the transfer function wrt. v1.
Frequency of oscillation in a Wien bridge (frequency selective circuit β):
$$ω_0=\frac{1}{CR}, f_0=\frac{1}{2\pi CR}$$
The final opamp is inverting (phase shift=180deg), therefore the 3 RC networks before it must create a 180deg phase shift (60deg each)
Transfer function of each CR network:
$$\frac{v_o}{v_i}(jω)=\frac{R}{R+\frac{1}{jωC}}=\frac{jωCR}{1+jωCR}$$
Frequency of oscillation in a phase shift oscillator (frequency selective circuit β): s.t. phase shift = 60deg
$$ω_0=\frac{1}{\sqrt{3}CR};\implies\angle60^{\circ}$$
Gainloss:
$$=[@ω_0=\frac{1}{\sqrt{3}CR}]|\frac{v_o}{v_i}|=\frac{1}{2};$$
3 stages means gainloss for whole circuit:
$$=(\frac{1}{2})^3=\frac{1}{8};|Aβ|=1\implies A=-8$$
Gain of final opamp:
$$A=\frac{R_2}{R}=8$$
R not frequency independent for phase shift oscillator; we can change R2 to ensure gain=(-)8.
